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Properties of galaxy dark matter halos from weak lensing

Properties of galaxy dark matter halos from weak lensing
Properties of galaxy dark matter halos from weak lensing

a r X i v :a s t r o -p h /0306515v 2 13 J a n 2004

Draft version February 2,2008

Preprint typeset using L A T E X style emulateapj v.14/09/00

PROPERTIES OF GALAXY DARK MATTER HALOS FROM WEAK LENSING

Henk Hoekstra 1,2,3,H.K.C.Yee 2,3,and Michael D.Gladders 3,4

Draft version February 2,2008

ABSTRACT

We present the results of a study of weak lensing by galaxies based on 45.5deg 2of R C band imaging data from the Red-Sequence Cluster Survey (RCS).We de?ne a sample of lenses with 19.5

We present the ?rst weak lensing detection of the ?attening of galaxy dark matter halos.We use a simple model in which the ellipticity of the halo is f times the observed ellipticity of the lens.We ?nd

a best ?t value of f =0.77+0.18

?0.21,suggesting that the dark matter halos are somewhat rounder than the light distribution.The fact that we detect a signi?cant ?attening implies that the halos are well aligned with the light distribution.Given the average ellipticity of the lenses,this implies a halo ellipticity of

e halo =0.33+0.07

?0.09,in fair agreement with results from numerical simulations of CDM.We note that this result is formally a lower limit to the ?attening,since the measurements imply a larger ?attening if the halos are not aligned with the light distribution.Alternative theories of gravity (without dark matter)predict an isotropic lensing signal,which is excluded with 99.5%con?dence.Hence,our results provide strong support for the existence of dark matter.

We also study the average mass pro?le around the lenses,using a maximum likelihood analysis.We consider two models for the halo mass pro?le:a truncated isothermal sphere (TIS)and an NFW pro?le.We adopt observationally motivated scaling relations between the lens luminosity and the velocity dispersion and the extent of the halo.The TIS model yields a best ?t velocity dispersion of σ=136±5±3km/s (all errors are 68%con?dence limits;the ?rst error bar indicates the statistical uncertainty,whereas

the second error bar indicates the systematic error)and a truncation radius s =185+30?28h

?1

kpc for a galaxy with a ?ducial luminosity of L B =1010h ?2

L B ⊙(under the assumption that the luminosity does not evolve with redshift).Alternatively,the best ?t NFW model yields a mass M 200=(8.4±0.7±

0.4)×1011h ?1M ⊙and a scale radius r s =16.2+3.6?2.9h

?1

kpc.This value for the scale radius is in excellent agreement with predictions from numerical simulations for a halo of this mass.

Subject headings:cosmology:observations ?dark matter ?gravitational lensing ?galaxies:haloes

1.introduction

The existence of massive dark matter halos around galaxies is widely accepted,based on di?erent lines of ev-idence,such as ?at rotation curves of spiral galaxies (e.g.,Van Albada &Sancisi 1986)and strong lensing systems (e.g.,Keeton,Kochanek &Falco 1998).However,rela-tively little is known about the properties of dark matter halos.Strong lensing only probes the gravitational poten-tial on small (projected)scales,whereas the lack of visi-ble tracers at large radii hamper dynamical methods.To date,only satellite galaxies have provided some informa-tion (e.g.,Zaritsky &White 1994;McKay et al.2002;Prada et al.2003).

A promising approach to study the galaxy dark matter halos is weak gravitational lensing.The tidal gravitational ?eld of the dark matter halo introduces small coherent dis-tortions in the images of distant background galaxies.The weak lensing signal can be measured out to large projected distances from the lens,and hence provides a unique

1CITA,University of Toronto,Toronto,Ontario M5S 3H8,Canada 2

Department of Astronomy,University of Toronto,Toronto,Ontario M5S 3H8,Canada

3Visiting Astronomer,Canada-France-Hawaii Telescope,which is operated by the National Research Council of Canada,Le Centre National de Recherche Scienti?que,and the University of Hawaii 4Observatories of the Carnegie Institution of Washington,813Santa Barbara Street,Pasadena,California 91101

probe of the gravitational potential on large scales.

The applications of this approach are numerous:one can infer masses of galaxies and compare the results to their luminosities (e.g.,McKay et al.2001;Wilson et al.2001),or one can attempt to constrain the halo mass pro-?le (e.g.,Brainerd et al.1996;Hudson et al.1998;Fischer et al.2000;Hoekstra et al.2003).Also,weak lensing can be used to constrain the shapes of halos by measuring the azimuthal variation of the lensing signal.Unfortunately,one can only study ensemble averaged properties,because the weak lensing signal induced by an individual galaxy is too low to be detected.

A successful measurement of the lensing signal requires large samples of both lenses and background galaxies.The ?rst attempt to detect the lensing signal by galaxies was made by Tyson et al.(1984)using photographic plates.It took more than a decade and CCD cameras before the ?rst detections were reported (Brainerd et al.1996;Grif-?ths et al.1996;Dell’Antonio &Tyson 1996;Hudson et al.1998).These early results were limited by the small areas covered by the observations.

The accuracy with which the galaxy-galaxy lensing sig-nal can be measured depends on the area of sky that is observed,and on the availabitity of redshifts for the lenses (as it allows for a proper scaling of the lensing signal).Photometric redshifts were used by Hudson et al.(1998)to scale the lensing signal of galaxies in the Hubble Deep 1

2Properties of dark matter halos

Field,and by Wilson et al.(2001)who measured the lens-ing signal around early type galaxies as a function of red-shift.Furthermore,several lensing studies targeted regions covered by redshift surveys.Smith et al.(2001)used790 lenses from the Las Campanas Redshift Survey;Hoekstra et al.(2003)used1125lenses from the Canadian Net-work for Observational Cosmology Field Galaxy Redshift Survey(CNOC2).The areas covered by these surveys are relatively small.

The Sloan Digital Sky Survey(SDSS)combines both survey area and redshift information.Its usefulness for galaxy-galaxy lensing was demonstrated clearly by Fischer et al.(2000).More recently,McKay et al.(2001)used the available SDSS redshift information to study the galaxy-galaxy lensing signal as a function of galaxy properties (also see Guzik&Seljak2002;Seljak2002).

The data used in this paper currently lacks redshift in-formation for the lenses.However,compared to previous work,the combination of large area and depth of our obser-vations allow us to measure the galaxy lensing signal with great precision.We use45.5deg2of R C-band imaging data from the Red-Sequence Cluster Survey(RCS).These data have been used previously for several weak lensing studies.Hoekstra et al.(2002a;2002b)placed joint con-straints on?m andσ8by measuring the lensing signal caused by large scale structure.Related to the subject of this paper is the study of the bias parameters as a func-tion of scale by Hoekstra et al.(2001b)and Hoekstra et al. (2002c).The latter studies made use of the galaxy-mass cross-correlation function measured from the RCS data. Here we use the galaxy-mass cross-correlation function for a di?erent purpose:we e?ectively deconvolve the cross-correlation function to study the properties of dark matter halos surrounding galaxies at intermediate redshifts.

The structure of the paper is as follows.In§2we brie?y discuss the data and the redshift distributions of the lenses and the sources.The ensemble averaged tangential shear around the lenses(galaxy-mass cross-correlation function) is presented in§3.In§4we use a maximum likelihood analysis to derive constraints on the extent of dark matter halos.The measurement of the projected shapes of the halos is presented in§5.

2.observations and analysis

We use the R C-band imaging data from the Red-Sequence Cluster Survey(Yee&Gladders2001;Gladders &Yee2003).The complete survey covers90deg2in both R C and z′,spread over22widely separated patches of ~2.1×2.3degrees.In this paper we use data from the northern half of the survey,which consists of10patches, observed with the CFH12k camera on the CFHT.These data cover45.5deg2on the sky,but because of masking the e?ective area is somewhat smaller.In the lensing anal-ysis we use a total of42deg2.A detailed description of the data reduction and object analysis can be found in Hoek-stra et al.(2002a),to which we refer for technical details. Here we present a short description of the various steps in the analysis.

We use single exposures in our analysis,and conse-quently cosmic rays have not been removed.However, cosmic rays are readily eliminated from the photometric catalogs:small,but very signi?cant objects are likely to be cosmic rays,or artifacts from the CCD.The object ?nder gives fair estimates of the object sizes,and we re-move all objects smaller than the size of the PSF.Some faint cosmic rays may hit galaxies,and consequently might not be recognized as cosmic rays.Based on the number of cosmic ray hits,and the area covered by galaxies we?nd that less than0.2%of the galaxies might be a?ected.Also, cosmic rays only introduce additional noise in the shape measurement,but do not bias the result.Consequently we conclude that remaining cosmic rays have a negligble e?ect on our results.

The objects in this cleaned catalog are then analysed, which yield estimates for the size,apparent magnitude, and shape parameters(polarisation and polarisabilities). The objects in this catalog are inspected by eye,in order to remove spurious detections.These objects have to be removed because their shape measurements are a?ected by cosmetic defects(such as dead columns,bleeding stars, halos,di?raction spikes)or because the objects are likely to be part of a resolved galaxy(e.g.,HII regions).

To measure the small,lensing induced distortions in the images of the faint galaxies it is important to accurately correct the shapes for observational e?ects,such as PSF anisotropy,seeing and camera shear;PSF anisotropy can mimic a cosmic shear signal,and a correction for the see-ing is required to relate the measured shapes to the real lensing signal.To do so,we follow the procedure outlined in Hoekstra et al.(1998).We select a sample of mod-erately bright stars from our observations,and use these to characterize the PSF anisotropy and seeing.We?t a second order polynomial to the shape parameters of the selected stars for each chip of the CFH12k camera.These results are used to correct the shapes of the galaxies for PSF anisotropy and seeing.

The e?ect of the PSF is not the only observational dis-tortion that has to be corrected.The optics of the camera stretches the images of galaxies(i.e.,it introduces a shear) because of the non-linear remapping of the sky onto the CCD.We have used observations of astrometric?elds to ?nd the mapping between the sky and the CCD pixel co-ordinate system,and derived the corresponding camera shear,which is subsequently subtracted from the galaxy ellipticity(see Hoekstra et al.1998).

The?ndings presented in Hoekstra et al.(2002a)sug-gest that the correction for PSF anisotropy has worked well.The absence of a“B”-mode on large scales in the measurements of the cosmic shear(Hoekstra et al.2002b) provides additional evidence that systematics are well un-der control(the small scale“B”-mode is attributed to intrinsic alignments).Furthermore,cosmic shear studies are much more sensitive to systematics than galaxy-galaxy lensing measurements(e.g.,see Hoekstra et al.2003).In galaxy-galaxy lensing one measures the lensing signal that is perpendicular to the lines connecting many lens-source pairs.These connecting lines are randomly oriented with respect to the PSF anisotropy,and hence suppress any residual systematics.

2.1.Redshift distributions

For the analysis presented here,we select a sample of “lenses”and“sources”on the basis of their apparent R C magnitude.We de?ne galaxies with19.5

Hoekstra et al.3

as lenses,and galaxies with21.5

For a singular isothermal sphere,the amplitude of the lensing signal depends on β ,the average ratio of the an-gular diameter distances between the lens and the source, D ls,and the distance between the observer and the source D s.More general,the signal also depends on D l,the dis-tance between the observer and the lens.Hence,to inter-pret the measurements,such as size and mass,one needs to know the redshift distributions of both lenses and sources. The CNOC2Field Galaxy Redshift Survey(e.g.,Lin et al.1999;Yee et al.2000;Carlberg et al.2001) has measured the redshift distribution of?eld galaxies down to R C=21.5,which is ideal,given our limits of 19.5

Compared to studies using SDSS data(McKay et al. 2001)we have the disadvantage that we do not have(spec-troscopic)redshifts for the individual lenses.As shown by Schneider&Rix(1997)and Hoekstra et al.(2003) this limits the accuracy of the measurements.We can de-rive useful constraints on the masses and extent of dark matter halos,but we have to assume scaling relations be-tween these parameters and the luminosity of the galaxies. Multi-color data for the northern part of the RCS will be available in the near future,allowing us to select a sample of lenses based on their photometric redshifts,and con-strain the scaling relations.

Nevertheless,the large area covered by the RCS allows us to derive interesting information about the properties of the lenses.In§4we use di?erent cuts in apparent mag-nitude to study the properties of dark matter halos using a maximum likelihood analysis.

For the source galaxies the situation is more compli-cated.These galaxies are generally too faint for spec-troscopic surveys,although recently Cohen et al.(2000) measured spectroscopic redshifts around the Hubble Deep Field North down to R C~24.Cohen et al.(2000)?nd that the spectroscopic redshifts agree well with the photometric redshifts derived from multi-color photome-try.Because of likely?eld-to-?eld variations in the red-shift distribution,we prefer to use the photometric red-shift distributions derived from both Hubble Deep Fields (Fern′a ndez-Soto et al.1999),which yields a median red-shift of z=0.53for the source galaxies.

The adopted source and lens redshift distributions result in an average value of β =0.29±0.01(average for the full sample of lenses and sources),where the error bar is based on the?eld-to-?eld variation and the?nite number of galaxies in the Hubble Deep Fields.The uncertainty in β a?ects predominantly our estimates for the galaxy masses(and velocity dispersions),but is negligible for the other model parameters.We note that for the galaxy-galaxy lensing analysis presented here the relevant param-eter is L0.3Bβ ,as opposed to simply β ,but we have veri?ed that results in a negligible change in the adopted systematic error.Throughout the paper we indicate the systematic error in the masses and velocity dispersions by a second error

bar.

Fig.1.—(a)The galaxy-mass cross-correlation function as a func-tion of angular scale.The lenses are selected on the basis on their apparent R C-band magnitude,taking19.5

3.galaxy-mass cross-correlation function The galaxy-mass cross-correlation function provides a convenient way to present the measurements.It is ob-tained from the data by measuring the tangential align-ment of the source galaxies with respect to the lens as a function of radius.Its use for studying the halos of galaxies is limited,because the clustering of galaxies complicates a direct interpretation of the signal:on small scales the signal is dominated by the mass distribution of the lens, but on larger scales one measures the superposition of the contributions from many lenses.

The observed galaxy-mass cross-correlation function as a function of angular scale is presented in Figure1a.A sig-ni?cant signal is detected out to one degree from the lens. If the signal presented in Figure1a is caused by gravita-tional lensing,no signal should be present when the phase of the distortion is increased byπ/4(i.e.,when the sources are rotated by45degrees).The results of this test,shown in Figure1b,suggest that residual systematics are negli-gible.

Before we can interpret the results we need to exam-ine the contribution of foreground galaxies.Some of the source galaxies will be in front of the lenses,and lower the lensing signal independent of radius.This is absorbed in

4Properties of dark matter halos

the value of β .These galaxies decrease the lensing sig-nal independent of angular scale.Some sources,however,are physically associated with the lenses.These galaxies cluster around the lenses,a?ecting the lensing signal more on small scales.We need to account for this source of contamination.To do so,we measure a fractional excess of sources around lenses which decreases with radius as f bg (r )=0.93r ?0.76(r in arcseconds),similar to what was found by Fischer et al.(2000).Under the assumption that the orientations of these galaxies are random (the tidal in-teraction with the lens has not introduced an additional tangential or radial alignment),the observed lensing signal has to be increased by a factor 1+f bg (r ).This assump-tion is supported by the ?ndings of Bernstein &Norberg (2002)who examined the tangential alignment of satellite galaxies around galaxies,extracted from the 2dF Galaxy Redshift Survey.The measurements presented in Figures 1have been corrected for this decrease in signal.We note that the correction for the presence of satellite galaxies is small,and has no signi?cant e?ect on our

results.

Fig. 2.—(a)Ensemble averaged tangential shear as a function of radius out to 2arcminutes from the lens.The solid line corre-sponds to the best ?t SIS model to the data at radii smaller than 2arcminutes.(b)The signal when the phase of the shear is increased by π/4.Note the di?erent vertical scale between panels a and b.

The signal on small angular scales is dominated by a single lens galaxy,and can be used to obtain an estimate of the mass weighted velocity dispersion of the sample of lenses (although such a mass estimate can still be slightly biased because of the clustering of the lenses).Figure 2a shows the ensemble averaged tangential shear on small scales.The measurements presented in Figure 2b show no evidence for residual systematics.

We ?t a singular isothermal sphere (SIS)model to the tangential shear at radii smaller than 2arcminutes (which corresponds to ~350h ?1kpc at the mean redshift of the lenses).The best ?t model is indicated by the dashed line in Figure 2a.For the Einstein radius r E we obtain a value of r E =0.′′140±0.′′009.If we extend the ?t to much larger

radii,the inferred value for the Einstein radius increases systematically.With our adopted redshift distributions for the lenses and the sources,the value of r E corresponds to a value of σ2 1/2=128±4km/s.The corresponding circular velocity can be obtained using V c =

Hoekstra et al.5 tributed in common halos,a simple interpretation of the

results becomes more di?cult.

Hoekstra et al.(2003)examined how known groups in

their observed?elds a?ected the lensing results,and found

that the masses and sizes might be overestimated by at

most~10%.Guzik&Seljak(2002)found similar results

from their analysis of the galaxy-galaxy lensing signal in

the context of halo models.Their approach allows one to

separate the contribution from groups to the lensing signal.

As expected,Guzik&Seljak(2002)found that the e?ect

depends on galaxy type:early type galaxies are found in

high density regions,and are a?ected more.Alternatively,

a comparison with numerical simulations which include a

prescription for galaxy formation(e.g.,Kau?mann et al.

1999a,1999b;Guzik&Seljak2001)can be used to quan-

tify this e?ect.

Another complication is the fact that we cannot sep-

arate the lenses in di?erent morphology classes with the

current data.Early and late type galaxies of a given lu-

minosity have di?erent masses,etc.(e.g.,Guzik&Seljak

2002).Hence,it is important to keep in mind that the re-

sults presented here are ensemble averages over all galaxy

types.This is where the SDSS can play an important role,

although we can signi?cantly improve the RCS results with

upcoming multi-color data.

The variance in the polarisations is approximately con-

stant with apparent magnitude and we approximate the

distribution by a Gaussian distribution.With the lat-

ter assumption,the log-likelihood follows aχ2distribution

with the number of degrees of freedom equal to the num-

ber of free model parameters,and the determination of

con?dence intervals is straightforward.The log-likelihood

is given by the sum over the two components of the polar-

isation e i of all the source galaxies

log L=? i,j e i,j?Pγj g model i,j2πGr2(r2+s2),(2)

whereσis the line-of-sight velocity dispersion,and s is a

truncation scale,i.e.the radius where the pro?le steepens.

On small scales(r?s)the model behaves as an Singular

Isothermal Sphere(SIS)model,whereas for r?s the den-

sity decreases∝r?4.The mass contained within a sphere

of radius r is given by

M(r)=

2σ2s

G

s=7.3×1012 σ1Mpc .(4)

The projected surface density for this model is given by

Σ(r)=σ2

6Properties of dark matter halos

relations,which allow us to relate the halo properties of the lenses to those of a ?ducial galaxy.For the ?ducial galaxy we take a luminosity of L B =1010h ?2L B ⊙.

Dynamical studies provide evidencence of a power law scaling relation between the velocity dispersion and the lu-minosity (e.g.,Tully-Fisher relation for spiral galaxies and Faber-Jackson relation for early type galaxies).We assume

σ∝L 0.3

B ,which is based on the observed slope of the B -band Tully-Fisher relation (e.g.,Verheijen 2001).Little is known,however,about the relation of the extent of dark matter halos with other (observable)https://www.doczj.com/doc/c317610808.html,ing

SDSS data,Guzik &Seljak (2002)?nd that M ∝L 1.2±0.2

g ′

.Motivated by this result,we adopt s ∝L 0.6

B ,which gives a total mass M ∝L 1.2B .We note,however,that we probe lower luminosities than Guzik &Seljak (2002)(also see McKay et al.2001)and as a result,the adopted scaling relation might di?er from the actual

ones.

Fig.3.—Joint constraints on the velocity dispersion σand trunca-tion parameter s for a ?ducial galaxy with L B =1010h ?2L B ⊙.The contours indicate the 68.3%,95.4%,and the 99.7%con?dence on two parameters jointly.The cross indicates the best ?t value.The dashed lines indicate models with masses M tot =10×1012h ?1M ⊙and M 200=5×1012h ?1M ⊙.

Figure 3shows the joint constraints on the velocity dis-persion σand truncation parameter s for a ?ducial galaxy

with L B =1010h ?2L B ⊙,under the assumption that the luminosity does not evolve with redshift.Conveniently,for this particular luminosity,the inferred values of the velocity dispersion and truncation parameter depend only marginally on the adopted scaling relations.

For the velocity dispersion we obtain a value of σ=136±5±3km/s (68%con?dence,marginalizing over all other model paremeters).Hoekstra et al.(2003)?nd a velocity dispersion of σ=110±12for a galaxy with a luminosity of 5.6×109h ?2L B ⊙.For a galaxy with that luminosity,the adopted scaling relations imply a velocity dispersion of 114±4±2km/s,in excellent agreement with Hoekstra et al.(2003).McKay et al.(2001)who found a best ?t value of σ=113+17?13km/s (95%con?dence)for

a galaxy with L g ′~9×109h ?2L g ′⊙.Our results corre-spond to a velocity dispersion of σ=127±5±3km/s for a galaxy of that luminosity,in good agreement with the SDSS result.

We derive tight constraints on the truncation param-eter,i.e.the extent of dark matter halos.We ?nd a

value of s =185+30?28h

?1

kpc (68%con?dence),and a to-tal mass M tot =(2.5±0.3±0.1)×1012h ?1M ⊙.The results presented in Hoekstra et al.(2003)imply a value

of s =290+139?82h

?1

kpc (68%con?dence)for their ?ducial galaxy.For a galaxy with a luminosity of 5.6×109h ?2L B ⊙,

we obtain s =131+21?20h ?1

kpc,marginally consistent with Hoekstra et al.(2003).

If we assume that L B ∝(1+z )we obtain a velocity dispersion of σ=146±5±3km/s and a truncation size

of s =213+35?32h

?1

kpc for a galaxy with L B (z =0)=1010h ?2

L B ⊙.

4.2.NFW model

Numerical simulations of collisionless cold dark matter (CDM)reproduce the observed structure in the universe remarkably well.Furthermore these simulations suggest that CDM gives rise to a speci?c density pro?le,which ?ts the radial mass distribution for halos with a wide range in mass (e.g.,Dubinski &Carlberg 1991;Navarro et al.1995,1996,1997).The NFW density pro?le is characterized by 2parameters,a density contrast δc and a scale r s

ρ(r )=

δc ρc

3

ρc r 3200,

(7)

with a corresponding rotation velocity V 200of

V 200=V (r 200)=

GM 200

3

c 3

Hoekstra et al.

7

test of the assumption that structures form through dissi-pationless collapse.The predicted pro?les agree well with the observed mass distribution in clusters of galaxies (e.g.,Hoekstra et al.2002d),but the situation is less clear for galaxy mass halos.Rotation curves can provide some con-straints,but typical values for r s for galaxy mass halos are 10?20h ?1kpc,comparable to the outermost point for which rotation curves have been measured.

Galaxies that are thought to be dark matter dominated,such as low surface brightness galaxies,potentially might be more suitable to test the CDM predictions.Studies of the rotation curves of low surface brightness galaxies suggest that,at least for a fraction of them,the observed rotation curves rise more slowly than the CDM predictions (e.g,de Blok,McGaugh &Rubin 2001;McGaugh,Barker &de Blok 2002).It is not clear,however,whether such studies provide a good test of CDM,because low surface brightness galaxies are peculiar (Zwaan &Briggs 2000),and their formation is not well understood.Hence,it is not obvious that their halos should be described by an NFW pro?le.Recently Ricotti (2003)has suggested that the inner slope might depend on halo mass,with low mass systems having shallow cores,whereas massive galaxies are well described by the NFW pro?le.

In this section we compare the NFW pro?le to the obser-vations,with δc and r s as free parameters.The equations describing the shear for the NFW pro?le have been derived by Bartelmann (1996)and Wright &Brainerd (2000).As before we have to adopt scaling relations.We as-sume that the maximum rotation velocity scales ∝L 0.3

B (the B-band Tully-Fisher relation;Verheijen 2001).If we

also assume that M 200∝L 1.2

B ,as motivated by the ?nd-ings of Guzik &Seljak (2002),we obtain that r s ∝L 0.75B

and δc ∝L ?0.85

B

.Figure 4shows the joint constraints on V 200and r s for a galaxy with a luminosity of L B =1010h ?2L B ⊙.In addi-tion,the right axis indicates the corresponding values for M 200.We derive a best ?t value of V 200=162±5±3km/s,or M 200=(8.4±0.7±0.4)×1011h ?1M ⊙(68%con?dence),

and a corresponding value of r 200=139+3?5h

?1

kpc.It is useful to compare this result with the mass from the TIS model.

The TIS model yields M TIS (r 200)=(1.0±0.1)×1012h ?1M ⊙,which is slightly larger than the NFW value.As shown by Wright &Brainerd (2000),isothermal mod-els give higher masses compared to NFW models.Hence,the results derived from both models are consistent.

From their galaxy-galaxy lensing analysis of the SDSS,Guzik &Seljak (2002)?nd M 200=(9.3±1.6)×1011h ?1M ⊙for a galaxy of L g ′~1.1×1010h ?2L g ′⊙,in good agreement with our results.

For the scale r s we ?nd r s =16.2+3.6?2.9h

?1

kpc (68%con?dence),and the best ?t density contrast is δc =

2.4+1.4?0.8×104

(68%con?dence;con?dence interval from Monte Carlo simulation).The TIS model provides a slightly better ?t to the data,but the di?erence is not signi?cant,and consequently the data are not su?cient to distinguish between the NFW and TIS model.

As before,we also calculated the results under the as-sumption that the luminosity evolves ∝(1+z ).In this

case we ?nd a value of r s =17.2+3.8?3.1h

?1

kpc and V 200=176±5±4km/s for a L B (z =0)=1010h ?2L B ⊙galaxy.

In our maximum likelihood analysis we considered r s and V 200(or equivalently the mass M 200)as free parame-ters.Numerical simulations of CDM,however,show that the parameters in the NFW model are correlated,albeit with some scatter.As a result,the NFW model can be considered as a one-parameter model:given the cosmol-ogy,redshift,and one of the NFW parameters,the values for all other parameters can be computed using the rou-tine CHARDEN made available by Julio Navarro 1.Hence,the simulations make a de?nite prediction for the value of V 200as a function of r s .The dotted line in Figure 4in-dicates this prediction.If the simulations provide a good description of dark matter halos,the dotted line should intersect our con?dence region,which it does.

This result provides important support for the CDM paradigm,as it predicts the correct “size”of dark matter halos.It is important to note that this analysis is a di-rect test of CDM (albeit not conclusive),because the weak lensing results are inferred from the gravitational poten-tial at large distances from the galaxy center,where dark matter dominates.Most other attempts to test CDM are con?ned to the inner regions,where baryons are,or might be,

important.

Fig.4.—Joint constraints on δc r 2s and scale radius r s

for a ?du-cial galaxy with L B =1010h ?2L B ⊙,with an NFW pro?le.The

contours indicate the 68.3%,95.4%,and the 99.7%con?dence on two parameters jointly.The cross indicates the best ?t value.The dashed lines indicate models with masses M 200=5×1011h ?1M ⊙and M 200=1012h ?1M ⊙.The dotted line indicates the predictions from the numerical simulations,which are in excellent agreement with our results.

5.shapes of halos

The average shape of dark matter halos can provide important information about the nature of dark matter.Numerical simulations of cold dark matter yield triaxial halos,with a typical ellipticity of ~0.3(e.g.,Dubinski &

1

The routine CHARDEN can be obtained from http://pinot.phys.uvic.ca/?jfn/charden

8Properties of dark matter halos

Carlberg1991).Hence,in the context of collisionless cold dark matter,the theoretical evidence for?attened halos is quite strong.If the dark matter is interacting,it tends to produce halos that are more spherical(compared to cold dark matter).This di?erence is more pronounced in the central parts of the halo,where the density is high.On the large scales probed by weak lensing,the di?erent types of dark matter(for reasonable interaction cross-sections) produces halos with similar shapes.

Nevertheless,a measurement of the average shape of dark matter halos is important,because the observational evidence is still limited.Dynamical measurements are limited by the lack of visible tracers,and therefore only probe the vertical potential on scales≤15kpc.Although the spread in inferred values for the axis ratio c/a(where c/a is the ratio of the shortest to longest principle axis of the halo)is large,the results suggest an average value of c/a=0.5±0.2(Sackett1999).

Weak gravitational lensing is potentially the most pow-erful way to derive constraints on the shapes of dark mat-ter halos.The amount of data required for such a mea-surement,however,is large(e.g.,Brainerd&Wright2000; Natarajan&Refregier2000):the galaxy-galaxy lensing signal is small,and now one needs to measure an even smaller azimuthal variation.We also have to assume that the galaxy and its halo are aligned.An imperfect align-ment between light and halo will reduce the amplitude of the azimuthal variation detectable in the weak lensing analysis.Hence,weak lensing formally provides a lower limit to the average halo ellipticity.

Brainerd&Wright(2000)and Natarajan&Refregier (2000)proposed to study the azimuthal variation in the tangential shear around the lenses.On very small scales, the lensing signal is dominated by the lens,but on larger scales,the clustering of the lenses will lower the signal one tries to measure(the two point function is axisymmetric). We therefore use the maximum likelihood approach used in the previous section.

To maximize the signal-to-noise ratio of the measure-ment one has to assign proper weights to the lenses:edge-on galaxies have maximal weight,whereas the lensing sig-nal around face-on galaxies contains no information about the shape of the halo.We adopt a simple approach, and assume that the(projected)ellipticity of the dark matter halo is proportional to the shape of the galaxy: e halo=fe lens.

The measurement of the azimuthally averaged tangen-tial shear around galaxies is robust against residual sys-tematics(e.g.,imperfect correction for PSF anisotropy): contributions from a constant or gradiant residual shear cancel.This is no longer the case for the quadrupole sig-nal,and imperfect correction for the PSF anisotropy can mimick the signal from a?attened halo.

If the lens galaxy is oriented randomly with respect to the residual shear,the average over many lenses will can-cel the contribution from systematics.In real data,how-ever,the uncorrected shapes of the lenses are aligned with the systematic signal.Hence,an imperfect correction can give rise to a small quadrupole signal,although we note that the lenses used in our analysis are large compared to the PSF.We estimate the amplitude of this e?ect in Ap-pendix A,and show that it is negligible for the measure-ments presented here.We also examined the robustness of our results by splitting the data into two samples and comparing the

results.

Fig.5.—?χ2as a function of f.We have assumed that the ellip-ticity of the halos is related to the observed ellipticity of the lens as e halo=fe lens.We have indicated the68.3%and95.4%con?dence

intervals.We?nd a best?t value of f=0.77+0.18

?0.21

(68%con?dence). Round halos(f=0)are excluded with99.5%con?dence.

We use an elliptical TIS model to compute the model shear?eld,and compare this to the data.Figure5shows the resulting?χ2as a function of f.We?nd a best?t

value of f=0.77+0.18

?0.21

(68%con?dence).This suggests that,on average,the dark matter distribution is rounder than the light distribution.As discussed above,our analy-sis formally provides only a lower limit on the halo elliptic-ity,and the true ellipticity might be higher if some of the halos are misaligned with the light.Nevertheless,the fact that we detect a signi?cant?attening implies that the ha-los are well aligned with the light distribution.Also note that the lensing signal is caused by a range of di?erent galaxy types,for which our simple relation between the halo ellipticity and light distribution might not be valid. Consequently the interpretation of the results is di?-cult,although a simple interpretation actually yields sen-sible results.For instance,the average ellipticity of the lens galaxies is e lens =0.414.Hence,the measured value of f implies an average projected halo ellipticity of

e halo =0.33+0.07

?0.09

(68%con?dence),which corresponds

to an projected axis ratio of c/a=0.67+0.09

?0.07

(68%con?-dence;where we have used c/a=1?e).Although the weak lensing yields a projected axis ratio,the result is in fair agreement with the results from numerical simulations.

A robust outcome of our analysis is that spherical ha-los(f=0)are excluded with99.5%con?dence.As we demonstrate below,this poses serious problems for alter-native theories of gravity,which attempt to explain the observations without dark matter.

Hoekstra et al.

9

5.1.Implications for alternative theories of gravity In this section we examine the implications of our mea-surement of the anisotropy in the lensing signal around galaxies for theories of gravity without dark matter.We focus on one particular approach:Modi?ed Newtonian Dynamics (MOND;Milgrom 1983;Sanders 1986;Sanders &McGaugh 2002),which has been shown to describe rota-tion curves rather well (e.g.,Begeman et al.1991;Sanders &Verheijen 1998).

In principle weak lensing can be used as a powerful test of MOND,but unfortunately no relativistic description of MOND has been found.Consequently one cannot com-pute the lensing signal in this theory.However,even in the absence of an appropriate description of lensing,we can use the observed anisotropy in the lensing signal around galaxies to test MOND.

In any reasonable alternative theory of gravity,the anisotropy in the lensing signal of an isolated galaxy is caused by the distribution of light and gas in that galaxy.In order to explain the ?at rotation curves,these alterna-tive theories typically need an e?ective force law ∝r ?1.Hence,on small scales one expects an anisotropic signal,but at large radii (where there are no stars and gas)we assume that the anisotropy in the lensing signal decreases ∝r ?2.As a result,these theories predict an almost isotropic weak lensing signal on the scales probed by our analysis,which is not observed.

Galaxies,however,are not isolated and the external ?eld e?ect (Milgrom 1986)might complicate the interpretation of our measurements.In MOND,if a galaxy is embed-ded in an external ?eld,this ?eld dominates the dynamics if its acceleration is larger than the acceleration of the galaxy.As a result,the e?ective gravitational ?eld is non-spherical,even if the potential of the galaxy is isotropic (as is the case for an isolated galaxy).

This e?ect is important if galaxies would be aligned with this external ?eld.We know,however,that the intrinsic alignments of galaxies are small (e.g.,Lee &Pen 2001,2002)and for the measurements presented here,it is safe to assume that the lenses have random orientation with respect to any external ?eld.Consequently,the observed anisotropy in the lensing signal cannot be caused by the external ?eld e?ect.

Hence,our ?ndings provide strong support for the exis-tence of dark matter,because alternative theories of grav-

ity predict an almost isotropic lensing signal.Better con-straints can be derived from future weak lensing surveys,which will allow us to study the anisotropy a function of projected distance from the galaxy.

6.conclusions

We have analysed the weak lensing signal caused by a sample of lenses with 19.5

The TIS model yields a best ?t velocity dispersion of σ=136±5±4km/s and a truncation radius s =

185+30?28h

?1

kpc for a galaxy with a ?ducial luminosity of L B =1010h ?2L B ⊙.Alternatively,the best ?t NFW model yields a mass M 200=(8.4±0.7±4)×1011h ?1M ⊙

and a scale radius r s =16.2+3.6?2.9h

?1

kpc.This value for the scale radius is in excellent agreement with predictions from numerical simulations for a halo of this mass.

We also present the ?rst detection of the ?attening of galaxy dark matter halos from weak lensing.We use a sim-ple model in which the ellipticity of the halo is f times the observed ellipticity of the lens.We ?nd a best ?t value

of f =0.77+0.18

?0.21(68%con?dence),suggesting that the dark matter halos are somewhat rounder than the light distribution.The fact that we detect a signi?cant ?at-tening implies that the halos are aligned with the light distribution.Given the average ellipticity of the lenses,

this implies a halo ellipticity of e halo =0.33+0.07

?0.09(68%con?dence),in fair agreement with results from numerical simulations of CDM.This result provides strong support for the existence of dark matter,as an isotropic lensing signal is excluded with 99.5%con?dence.

We are grateful to the anonymous referee whose com-ments have signi?cantly improved the quality of this pa-per.The RCS project is partially supported by grants from the Natural Science and Engineering Science Coun-cil of Canada and the University of Toronto to HKCY.

APPENDIX

contribution of systematics to the shape measurement of dark halos

In this appendix we examine how residual systematics a?ect the measurement of the ?attening of dark matter halos.A schematic overview of the situation is presented in Figure A6.The thin lines indicate the direction of residual systematics.The residual shear has an amplitude ?γand a position angle φwith respect to the major axis of the lens.The tangential

shear γobs T observed at a position (r,θ)is the sum of the lensing signal γlens

T and the contribution from systematics ?γT .The latter is given by

?γT =??γ[cos(2φ)cos(2θ)+sin(2φ)sin(2θ)]=??γcos(2(θ?φ)).(A1)

Hence,the azimuthally averaged tangential shear is not a?ected by systematics as

dθ?γT (θ)=0,and the weak lensing

mass estimate is very robust.For a ?attened halo,the lensing signal γlens

T is given by

γlens

T (r,θ)=[1+γf cos(2θ)]· γT (r ),

(A2)

10Properties of dark matter halos

where γT is the azimuthally averaged tangential shear,andγf is a measure of the?attening of the halo.For positive values ofγf,the halo is aligned with the lens.

We consider the worst case scenario,and demonstrate that even in this situation the results are robust.One way to estimate the?attening of the halo is to measure the shearsγ+(atθ=0andπ),andγ?(atθ=π/2and3π/2).The observed ratio f=γ?/γ+is

γ?+?γcos(2φ)

f obs=

γ21+γ22,we?nd

cos(2φ) =α?γ

π

.(A5)

This introduces a relatively large signal because face-on lenses(γ≈0)align easily with the PSF anisotropy.Such lenses,however,contain no information about the shape of the halo.In our analysis we assign a weight∝γto each lens. Hence in our case we are less sensitive to systematics as the correct estimate is given by

cos(2φ) =α?γ

Hoekstra et al.11 Hence,the observed ratioγ?/γ+reduces to

γ?+?γ2α/2

f obs=

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入门指南。如果您没有经验,您可以在该向导的指引下完成基本的操作和设置。我们建议您熟悉后, 关闭操作入门指南,仔细阅读说明书,熟悉和掌握软件的操作。 “关闭入门指南”后,操作界面如下。 1.2设备参数设置 1.3部门和注册卡用户操作 1.3.1设置部门 点击【设置】>【部门】,进入部门界面。 点击【添加最高级部门】。

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(此文档为word格式,可任意修改编辑!)

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图表目录 图表1:2016年国内汽车产销量均突破2800万辆 (4) 图表2:2017Q1,全国汽车保有量突破2亿辆 (5) 图表3:我国千人汽车保有量低于世界平均水平,远低于发达国家 (6) 图表4:汽车注销、报废、拆解的流转过程 (7) 图表5:国内报废汽车回收拆解行业发展稳步推进 (9) 图表6:2014年起开始报废汽车回收量明显增加 (10) 图表7:近年国内的汽车回收量稳中有升,实际回收率依旧偏低11 图表8:近年我国关于汽车拆解出台的系列政策 (12) 图表9:正规渠道回收车辆占比只有近一半 (13) 图表10:车辆拆解产值中钢铁占比最大,7成左右 (13) 图表11:中国黄标车数量2013年约有1300万辆 (15) 图表12:中国汽车注销比例远低于发达国家 (15) 图表13:预计2020年汽车回收行业产值将约千亿元 (16) 图表14:国内报废汽车回收拆解行业的相关上市公司 (18)

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————————————————————————————————作者:————————————————————————————————日期: 2

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